Multivariable calculus results in different countries

Abstract

In this paper we present the results of a survey of research studies on the learning of multivariable calculus in a wide geographic spectrum. The goal of this study is to describe what research results tell us about students’ learning of calculus of two-variable functions, and how they inform its teaching. In spite of the diversity of cultures and theoretical approaches, results obtained are coherent and similar in terms of students’ learning, and of teaching strategies designed to help students understand two-variable calculus deeply. Results show the need to introduce students to the geometry of three-dimensional space and vectors, the importance of the use of graphics and geometrical representations, and of thorough work on functions before introducing other calculus topics. The reviewed research deals in different ways with the idea of generalization concerning how students’ knowledge of one-variable calculus influences their understanding of multivariable calculus. Some research-based teaching strategies that have been experimentally tested with good results are included. Results discussed include the following: basic aspects of functions of two variables, limits, differential calculus, and integral calculus. The survey shows that there is still a need for research using different theoretical perspectives, on the transition from one-variable calculus to two-variable and multivariable calculus.

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Notes

  1. 1.

    The authors thank a referee for suggesting notions of ‘open statement’ and ‘presumed context’.

  2. 2.

    fx denotes \(\frac{\partial f}{\partial x}\).

References

  1. Alves, F. R. V. (2011). Aplicações da Sequência Fedathi na promoção das categorias do raciocínio intuitivo no Cálculo a Várias Variáveis (Unpublished doctoral dissertation). Universidade Federal do Ceará, Fortaleza, Brazil

  2. Alves, F. R. V. (2014). Análise e visualização de formas quadráticas na determinação de pontos extremante. Revista Eletrônica Debates em Educação Científica e Tecnológica, 4(1), 128–149.

    Google Scholar 

  3. Bajracharya, R. R., Emigh, P. J., & Manogue, C. A. (2019). Students’ strategies for solving a multirepresentational partial derivative problem in thermodynamics. Physical Review Physics Education Research, 15(2), 020124.

    Article  Google Scholar 

  4. Dorko, A. (2017). Students’ generalization of function from single- to multivariable settings (Unpublished doctoral dissertation). Oregon State University, USA

  5. Dorko, A., & Speer, N. (2013). Calculus students’ understanding of volume. Investigations in Mathematics Learning, 6(2), 48–68.

    Article  Google Scholar 

  6. Dorko, A., & Weber, E. (2014). Generalizing calculus ideas from two dimensions to three: How multivariable calculus students think about domain and range. Research in Mathematics Education, 16(3), 269–287.

    Article  Google Scholar 

  7. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131.

    Article  Google Scholar 

  8. Fisher, B. (2008). Students’ conceptualizations of multivariable limits (Unpublished doctoral dissertation). Oklahoma State University.

  9. Gemechu, E., Kassa, M., & Atnafu, M. (2018). Matlab supported learning and students’ conceptual understanding of functions of two variables: Experiences from Wolkite university. Bulgarian Journal of Science and Education Policy (BJSEP), 12(2), 314–344.

    Google Scholar 

  10. Harel, G., & Tall, D. (1989). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38–42.

    Google Scholar 

  11. Henriques, A. (2006). L’enseignement et l’apprentissage des integrales multiples: Analyse didactique intégrant l’usage du logiciel Maple (Unpublished doctoral dissertation). Université Joseph Fourier, France.

  12. Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. Journal of Mathematical Behavior, 32, 122–141.

    Article  Google Scholar 

  13. Jones, S. R. (2020). Scalar and vector line integrals: A conceptual analysis and an initial investigation of student understanding. Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2020.100801.

    Article  Google Scholar 

  14. Jones, S. R., & Dorko, A. (2015). Students’ understandings of multivariate integrals and how they may be generalized from single integral conceptions. Journal of Mathematical Behavior, 40, 154–170.

    Article  Google Scholar 

  15. Kabael, T. (2011). Generalizing single variable functions to two-variable functions, function machine and APOS. Educational Sciences: Theory and Practice, 11(1), 484–499.

    Google Scholar 

  16. Kashefi, H., Zaleha, I., & Yudariah, M. Y. (2010). Obstacles in the learning of two-variable functions through mathematical thinking approach. Procedia Social and Behavioral Sciences, 8, 173–180.

    Article  Google Scholar 

  17. Kashefi, H., Zaleha, I., & Yudariah, M. Y. (2013). Learning functions of two variables based on mathematical thinking approach. Jurnal Teknologi, 63(2), 59–69.

    Article  Google Scholar 

  18. Kashefi, H., Zaleha, I., Yudariah, M. Y., & Roselainy, A. R. (2012). Supporting students mathematical thinking in the learning of two-variable functions through blended learning. Procedia Social and Behavioral Sciences, 46, 3689–3695.

    Article  Google Scholar 

  19. Kustusch, M. B., Roundy, D., Dray, T., & Manogue, C. A. (2014). (2014) Partial derivative games in thermodynamics: A cognitive task analysis. Physical Review Physics Education Research, 10, 010101.

    Article  Google Scholar 

  20. Mamona-Downs, J. K., & Megalou, F. J. (2013). Students’ understanding of limiting behavior at a point for functions from R2 to R. Journal of Mathematical Behavior, 32, 53–68.

    Article  Google Scholar 

  21. Martínez-Planell, R., & Trigueros, M. (2012). Students’ understanding of the general notion of a function of two variables. Educational Studies in Mathematics, 81(3), 365–384.

    Article  Google Scholar 

  22. Martínez-Planell, R., & Trigueros, M. (2013). Graphs of functions of two variables: Results from the design of instruction. International Journal of Mathematical Education in Science and Technology, 44(5), 663–672.

    Article  Google Scholar 

  23. Martínez-Planell, R., & Trigueros, M. (2019). Using cycles of research in APOS: The case of functions of two variables. The Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2019.01.003.

    Article  Google Scholar 

  24. Martínez-Planell, R., & Trigueros, M. (2020). Students’ understanding of Riemann sums for integrals of functions of two variables. The Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2020.100791.

    Article  Google Scholar 

  25. Martínez-Planell, R., Trigueros, M., & McGee, D. (2015). On students’ understanding of the differential calculus of functions of two variables. Journal of Mathematical Behavior, 38, 57–86.

    Article  Google Scholar 

  26. Martínez-Planell, R., Trigueros, M., & McGee, D. (2017). Students’ understanding of the relation between tangent plane and directional derivatives of functions of two variables. The Journal of Mathematical Behavior, 46, 13–41.

    Article  Google Scholar 

  27. McGee, D., & Martínez-Planell, R. (2014). A study of semiotic registers in the development of the definite integral of functions of two and three variables. International Journal of Science and Mathematics Education, 12(4), 883–916.

    Article  Google Scholar 

  28. McGee, D., & Moore-Russo, D. (2015). Impact of explicit presentation of slopes in three dimensions on students’ understanding of derivatives in multivariable calculus. International Journal of Science and Mathematics Education, 13(Suppl 2), 357–384.

    Article  Google Scholar 

  29. McGee, D., Moore-Russo, D., Ebersole, D., Lomen, D., & Marin, M. (2012). Using physical manipulatives in the multivariable calculus classroom. PRIMUS, 22(4), 265–283.

    Article  Google Scholar 

  30. Megalou, F. (2018). Students’ choices of definitions for the concept of limit of functions from R2 to R. International Journal of Mathematical Education in Science and Technology, 50(1), 141–156.

    Article  Google Scholar 

  31. Moore-Russo, D., Conner, A., & Rugg, K. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3–21.

    Article  Google Scholar 

  32. Moreno-Arotzena, O., Pombar-Hospitaler, I., & Barraguéz, J. I. (2020). University student understanding of the gradient of a function of two variables: An approach from the perspective of the theory of semiotic representation registers. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-020-09994-9.

    Article  Google Scholar 

  33. National Research Council. (2006). Learning to think spatially: GIS as a support system in the K–12 curriculum. Washington, DC: National Academic Press.

    Google Scholar 

  34. Newcombe, N. S. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator, 34, 29–35.

    Google Scholar 

  35. Nguyen, D., & Rebello, N. S. (2011). Students’ difficulties with integration in electricity. Physics Education Research Physical Review Special Topics. https://doi.org/10.1103/PhysRevSTPER.7.010113.

    Article  Google Scholar 

  36. Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics (pp. 27–42). Washington, DC: Mathematical Association of America.

    Google Scholar 

  37. Pittalis, M., & Christou, C. (2010). Types of reasoning in 3D geometry thinking and their relation with spatial ability. Educational Studies in Mathematics, 75(2), 191–212.

    Article  Google Scholar 

  38. Roundy, D., Kustusch, M. B., & Manogue, C. (2014). Name the experiment! Interpreting thermodynamic derivatives as thought experiments. American Journal of Physics, 82, 39.

    Article  Google Scholar 

  39. Sandoval, I., & Possani, E. (2016). An analysis of different representations for vectors and planes in R3. Educational Studies in Mathematics, 92, 109–127.

    Article  Google Scholar 

  40. Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245.

    Article  Google Scholar 

  41. Şefik, Ö., & Dost, Ş. (2020). The analysis of the understanding of three dimensional Euclidian space and the two-variable function concept by university students. The Journal of Mathematical Behavior, 57, 100697. https://doi.org/10.1016/j.jmathb.2019.03.004he.

    Article  Google Scholar 

  42. Stewart, J. (2006). Calculus: Early transcendentals, 6E. Pacific Grove: Thompson Brooks/Cole.

    Google Scholar 

  43. Tall, D. (1992a). Visualizing differentials in two and three dimensions. Teaching Mathematics and its Applications, 11(1), 1–7.

    Article  Google Scholar 

  44. Tall, D. (1992b). Conceptual foundations of the calculus and the computer. In Proceedings of the fourth annual international conference on technology in collegiate mathematics (pp. 73–88). New York: Addison-Wesley.

  45. Thompson, J. R., Bucy, B. R., & Mountcastle, D. B. (2006). Assessing student understanding of partial derivatives in thermodynamics. Proceedings of the American Institute of Physics, 818, 77–80.

    Google Scholar 

  46. Trigueros, M., & Martínez-Planell, R. (2010). Geometrical representations in the learning of two variable functions. Educational Studies in Mathematics, 73(1), 3–19.

    Article  Google Scholar 

  47. Trigueros, M., Martínez-Planell, R., & McGee, D. (2018). Student understanding of the relation between tangent plane and the total differential of two-variable functions. International Journal of Research in Undergraduate Mathematics Education, 4(1), 181–197.

    Article  Google Scholar 

  48. Vigo Ingar, K. (2014). A Visualização na Aprendizagem dos Valores Máximos e Mínimos Locais da Função de Duas Variáveis Reais (Unpublished doctoral dissertation). Pontifícia Universidade Católica de São Paulo, Brazil.

  49. Wangberg, A., & Johnson, B. (2013). Discovering calculus on the surface. PRIMUS, 23(7), 627–639.

    Article  Google Scholar 

  50. Weber, E. D. (2012). Students’ ways of thinking about two-variable functions and rate of change in space (Unpublished doctoral dissertation). Arizona State University, USA.

  51. Weber, E. D. (2015). The two-change problem and calculus students’ thinking about direction and path. The Journal of Mathematical Behavior, 37, 83–93.

    Article  Google Scholar 

  52. Weber, E., Tallman, M., Byerley, C., & Thompson, P. W. (2012). Introducing derivative via the calculus triangle. Mathematics Teacher, 104(4), 274–278.

    Article  Google Scholar 

  53. Weber, E., & Thompson, P. W. (2014). Students’ images of two-variable functions and their graphs. Educational Studies in Mathematics, 87(1), 67–85.

    Article  Google Scholar 

  54. Williams, S. (1991). Models of limits held by college calculus students. Journal of Research in Mathematics Education, 22(3), 219–236.

    Article  Google Scholar 

  55. Xhonneux, S. (2011). Regard institutionnel sur la transposition didactique du Théorème de Lagrange en mathématique et en économie (Unpublished doctoral dissertation). Université de Namur, Belgium.

  56. Yerushalmy, M. (1997). Designing representations: Reasoning about functions of two variables. Journal of Research in Mathematics Education, 28, 431–466.

    Article  Google Scholar 

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Acknowledgements

This project was partially funded by Asociación Mexicana de Cultura A.C.

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Correspondence to Rafael Martínez-Planell.

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Martínez-Planell, R., Trigueros, M. Multivariable calculus results in different countries. ZDM Mathematics Education (2021). https://doi.org/10.1007/s11858-021-01233-6

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Keywords

  • Multivariable calculus
  • Functions of two variables
  • Differential calculus
  • Integral calculus